324 
Proceedings of Royal Society of Edinburgh. [sess. 
let us first establish a few simple cases, in order that the 
principle at the basis of the demonstration may become familiar, 
and the law of formation of the quotient come gradually into 
evidence. ^ 
The simplest of all cases is where the two terms differ in only 
two of the elements, — for example, the terms A^B2C3D^E5 and 
A1B2C3D5E4. Here it is evident that 
A1B2C3D4E5 — AJB2C3D5E4 = A3^B2C3*1D^E5|. 
But ID4E5I, as a minor of the adjugate and of the 2nd order, is 
equal to A multiplied by the complementary minor of the corre- 
sponding minor in the original determinant. Hence 
Next, taking the case where three of the elements differ, we 
have 
^1^2^3^4^5 ~ ^1^2^4^5^3 ~ -^1^2^3^4^5 “ ^1^2^3^5®4 
-f AJB2C3D5E4 — Aj^B2C4Df^E3 , 
= A,B2C3 -|D,E,| - AiB2 D,.1C3EJ, 
= { Aj^B2C3*ja]^&2'^3l - AjB 2D5-| aj&2^5l} ■ ^ • 
And now, as if quite generally, let us take the terms A5BJC4D3E2 
' and A2B4C1D5E3. Directing our attention to the interchanges 
which must take place between indices of the former in order to 
transform it into the latter, we see that the intermediate stages of 
transformation may be 
A2B4C4D3E3. 
Affixing each of these, first with the negative sign and then with the 
positive sign, to the difference in question, we have 
A5B1C4D3E, - A2B4CiD5E3= A5B1C4D3E0 - A5B4C4D3E2 
+ A5B4C1D3E0 - A2 B4C4D3E5 
-I- A2B4C4D3E5 - A2B4C4D5E3, 
= AgD3E3 -| B4C4 I - B4CJD3 -I AgEg) + A2B4C1 •) D3Eg ), 
= I A5T)3E2* I 1 — B4C4D3' ) | + A2B4C4’ | , A . 
